Chegg A manufacturer of submersible pumps claims that at most 30% of the pumps require repairs within the first 5 years of operations. A random sample of 120 of these pumps included 47 which required repairs within the first 5 years. Does the manufacturer’s claim appear to be true, at the 0.05 level of significance?

We are given that a manufacturer of submersible pumps claims that at most 30% of the pumps require repairs within the first 5 years of operations. A random sample of 120 of these pumps included 47 which required repairs within the first 5 years.

We have to test the manufacturer’s claim.

Firstly, as we know that the testing is done always on the population parameter.

Let p = % of the pumps that require repairs within the first 5 years of operations.

SO, Null Hypothesis, [tex]H_0[/tex] : [tex]p \leq[/tex] 30% {means that at most 30% of the pumps require repairs within the first 5 years of operations}

Alternate Hypothesis, [tex]H_a[/tex] : p > 30% {means that more than 30% of the pumps require repairs within the first 5 years of operations}

The test statistics that will be used here is One-sample z proportion test;

T.S. = [tex]\frac{\hat p-p}{\sqrt{\frac{\hat p(1-\hat p)}{n} } }[/tex] ~ N(0,1)

where, [tex]\hat p[/tex] = proportion of the pumps that require repairs within the first 5 years of operations in a sample of 120 = [tex]\frac{47}{120}[/tex]

n = sample of pumps 120

So, test statistics = [tex]\frac{\frac{47}{120} -0.30}{\sqrt{\frac{\frac{47}{120}(1-\frac{47}{120})}{120} } }[/tex]

= 2.057

Now, at 0.05 level of significance, the z table gives critical value of 1.6449. Since our test statistics is more than the critical value of z so we have sufficient evidence to reject our null hypothesis as it will fall in the rejection region.

Therefore, we conclude that more than 30% of the pumps require repairs within the first 5 years of operations which means the manufacturer’s claim was not true.